The following article is reprinted from the October/September, 2001 issue
of On the Edge, the Interactive Data Fixed Income Analytics bimonthly newsletter.
Volatility Measures in BondEdge
Teri Geske
Senior Vice President, Product Development
Volatility is an important input to fixed income models, affecting the
effective duration, convexity, option values and OAS for securities and
derivatives with embedded options (such as callable bonds, mortgage-backed
securities, caps and floors, etc.). This article discusses the specific
way volatility rates are used in BondEdge’s models, with some background
on how volatility rates are estimated and why these volatilities can differ
across modeling sources.
First, it is worth reviewing what is meant by “volatility”
with respect to modeling fixed income securities. Volatility refers
to the annualized standard deviation of the change in the interest rate(s)
that affect the value of securities or derivatives with interest rate
sensitive cash flows. So, a volatility of 10% would imply that when
modeling possible future interest rate environments over a one-year
horizon, approximately 2/3rds of the outcomes would be within +/- 10%
of current levels. Of course, the term “interest rates” is
rather generic – are we referring to short-term interest rates,
long term rates, or all rates? Some models apply the same volatility
across the entire term structure. Other models employ a “term structure”
of volatility, meaning that different volatilities are associated with
different points along the curve. In this case, a “short rate volatility”
determines the standard deviation of the movement in short-term interest
rates, a “long rate volatility” determines the standard deviation
of long-term interest rates, and so on. So, in comparing models it is
important to know whether one or more volatility rates are employed.
A “term structure” of volatility is a desirable feature,
as empirically there is evidence that short-term rates are more volatile
than long rates, and using different volatilities along the yield curve
allows a model to generate a richer set of yield curves than one that
uses a single volatility parameter. In constructing a term structure
of volatility, a financial modeler could choose to explicitly specify
separate volatility rates for each point on the yield curve; for example,
a model using a 9-point yield curve could have nine distinct volatility
rates as inputs. While this is theoretically possible to do, such flexibility
would make the model computationally slower than one that applies a
single rate volatility across the entire yield curve. So, it would be
nice to find some compromise between the two approaches, which is what
we have chosen to do in BondEdge.
The Heath-Jarrow-Morton (HJM) term structure model used in BondEdge
constructs an implied term structure of volatility based on a long-rate
volatility and a mean-reversion factor. These inputs, when applied to
the initial (current) yield curve, produce different volatilities for
the short, intermediate and long maturity points on the yield curve.
So, BondEdge uses a short rate volatility to determine the dispersion
of short-term interest rates within the Monte Carlo analytics and corporate
bond option model. The HJM framework also computes a separate volatility
for the 10-year Treasury rate, a key input to the refinancing component
of fixed rate mortgage prepayments. In a normal (upward-sloping) yield
curve environment, short and intermediate rate volatilities in BondEdge
are higher than the long-rate volatility (how much higher depends on
the level and slope of the yield curve). This is consistent with historical
observations (and with economic intuition) that short-term rates are
typically more volatile than long-term rates (on a proportional basis).
So, while BondEdge does not explicitly accept volatility inputs for
each point along the yield curve, the system does use a “term structure”
of volatility. You can graph BondEdge’s term structure of volatility
in the Volatility Appraisal report, accessed via the Simulation menu,
and can determine the impact of changing the long or short rate volatilities,
or the mean-reversion factor, in this Simulation or in the Security
Valuation analysis.
How do we determine the volatility rates used in BondEdge? In general,
volatilities can be estimated in one of two ways (note that there are
sophisticated refinements of these techniques, and they can even be
combined). Volatilities can be computed directly from historical data,
i.e., derived from observing the actual volatility of interest rates
at each point on the curve over some period of time. Alternatively,
they can be implied from current prices of derivative instruments. For
example, since interest rate caps are options on interest rate moves,
the volatility rate used to price a cap maturing N years in the future
is the market’s implied forecast of interest rate volatility for
time N on the yield curve. Both the historical and implied approaches
have merit but they are unlikely to produce the same result (e.g., today’s
implied 5-year volatility is not necessarily a good predictor of how
volatile the 5 year Treasury yield will actually be over the next 6
or 12 months). So, the choice of which volatility rates to use is somewhat
subjective and is another reason volatilities can vary from one modeling
source to another. Volatilities implied from derivatives prices are
often used in a trader’s model, because traders manage positions
over a short time horizon (they are looking to capture profits from
market moves over a few days, or even intraday), and implied volatilities
give the best indication of where the market thinks volatilities are
today. In BondEdge, the inputs that determine the term structure of
volatility are based on data, primarily because a portfolio manager’s
time horizon is typically measured over a fairly long time period (months,
not days), and implied volatilities tend to be higher than what historical
volatilities turn out to be, ex post.
In summary, volatility rates can be the source of differences when
comparing option-adjusted measures from different systems. BondEdge
derives a term structure of volatility that typically causes the volatility
of short-term rates to be higher than the long rate volatility; these
volatilities determine the dispersion of yield curve shifts that are
generated within the Monte Carlo analytics and corporate bond option
model. We hope this review has been helpful – if you have further
questions on this topic, please contact your Interactive Data Fixed Income Analytics representative.
